the earth was derived from reasoning on the effects of the earth's
_centrifugal force_, occasioned by its rapid revolution on its own axis.
When water is whirled in a pail, we see it recede from the centre and
accumulate upon the sides of the vessel; and when a millstone is whirled
rapidly, since the portions of the stone furthest from the centre
revolve much more rapidly than those near to it, their greater tendency
to recede sometimes makes them fly off with a violent explosion. A case,
which comes still nearer to that of the earth, is exhibited by a mass of
clay revolving on a potter's wheel, as seen in the process of making
earthen vessels. The mass swells out in the middle, in consequence of
the centrifugal force exerted upon it by a rapid motion. Now, in the
diurnal revolution, the equatorial parts of the earth move at the rate
of about one thousand miles per hour, while the poles do not move at
all; and since, as we take points at successive distances from the
equator towards the pole, the rate at which these points move grows
constantly less and less; and since, in revolving bodies, the
centrifugal force is proportioned to the velocity, consequently, those
parts which move with the greatest rapidity will be more affected by
this force than those which move more slowly. Hence, the equatorial
regions must be higher from the centre than the polar regions; for, were
not this the case, the waters on the surface of the earth would be
thrown towards the equator, and be piled up there, just as water is
accumulated on the sides of a pail when made to revolve rapidly.
Huyghens, an eminent astronomer of Holland, who investigated the laws of
centrifugal forces, was the first to infer that such must be the actual
shape of the earth; but to Sir Isaac Newton we owe the full developement
of this doctrine. By combining the reasoning derived from the known laws
of the centrifugal force with arguments derived from the principles of
universal gravitation, he concluded that the distance through the earth,
in the direction of the equator, is greater than that in the direction
of the poles. He estimated the difference to be about thirty-four miles.
But it was soon afterwards determined by the astronomers of France, to
ascertain the figure of the earth by actual measurements, specially
instituted for that purpose. Let us see how this could be effected. If
we set out at the equator and travel towards the pole, it is easy to see
when we have advanced one degree of latitude, for this will be indicated
by the rising of the north star, which appears in the horizon when the
spectator stands on the equator, but rises in the same proportion as he
recedes from the equator, until, on reaching the pole, the north star
would be seen directly over head. Now, were the earth a perfect sphere,
the meridian of the earth would be a perfect circle, and the distance
between any two places, differing one degree in latitude, would be
exactly equal to the distance between any other two places, differing in
latitude to the same amount. But if the earth be a spheroid, flattened
at the poles, then a line encompassing the earth from north to south,
constituting the terrestrial meridian, would not be a perfect circle,
but an ellipse or oval, having its longer diameter through the equator,
and its shorter through the poles. The part of this curve included
between two radii, drawn from the centre of the earth to the celestial
meridian, at angles one degree asunder, would be greater in the polar
than in the equatorial region; that is, the degrees of the meridian
would lengthen towards the poles.
The French astronomers, therefore, undertook to ascertain by actual
measurements of arcs of the meridian, in different latitudes, whether
the degrees of the meridian are of uniform length, or, if not, in what
manner they differ from each other. After several indecisive
measurements of an arc of the meridian in France, it was determined to
effect simultaneous measurements of arcs of the meridian near the
equator, and as near as possible to the north pole, presuming that if
degrees of the meridian, in different latitudes, are really of different
lengths, they will differ most in points most distant from each other.
Accordingly, in 1735, the French Academy, aided by the government, sent
out two expeditions, one to Peru and the other to Lapland. Three
distinguished mathematicians, Bouguer, La Condamine, and Godin, were
despatched to the former place, and four others, Maupertius, Camus,
Clairault, and Lemonier, were sent to the part of Swedish Lapland which
lies at the head of the Gulf of Tornea, the northern arm of the Baltic.
This commission completed its operations several years sooner than the
other, which met with greater difficulties in the way of their
enterprise. Still, the northern detachment had great obstacles to
contend with, arising particularly from the extreme length and severity
of their Winters. The measurements, however, were conducted with care
and skill, and the result, when compared with that obtained for the
length of a degree in France, plainly indicated, by its greater amount,
a compression of the earth towards the poles.
Mean-while, Bouguer and his party were prosecuting a similar work in
Peru, under extraordinary difficulties. These were caused, partly by the
localities, and partly by the ill-will and indolence of the inhabitants.
The place selected for their operations was in an elevated valley
between two principal chains of the Andes. The lowest point of their arc
was at an elevation of a mile and a half above the level of the sea;
and, in some instances, the heights of two neighboring signals differed
more than a mile. Encamped upon lofty mountains, they had to struggle
against storms, cold, and privations of every description, while the
invincible indifference of the Indians, they were forced to employ, was
not to be shaken by the fear of punishment or the hope of reward. Yet,
by patience and ingenuity, they overcame all obstacles, and executed
with great accuracy one of the most important operations, of this
nature, ever undertaken. To accomplish this, however, took them nine
years; of which, three were occupied in determining the latitudes
alone.[5]
I have recited the foregoing facts, in order to give you some idea of
the unwearied pains which astronomers have taken to ascertain the exact
figure of the earth. You will find, indeed, that all their labors are
characterized by the same love of accuracy. Years of toilsome watchings,
and incredible labor of computation, have been undergone, for the sake
of arriving only a few seconds nearer to the truth.
The length of a degree of the meridian, as measured in Peru, was less
than that before determined in France, and of course less than that of
Lapland; so that the spheroidal figure of the earth appeared now to be
ascertained by actual measurement. Still, these measures were too few in
number, and covered too small a portion of the whole quadrant from the
equator to the pole, to enable astronomers to ascertain the exact law of
curvature of the meridian, and therefore similar measurements have since
been prosecuted with great zeal by different nations, particularly by
the French and English. In 1764, two English mathematicians of great
eminence, Mason and Dixon, undertook the measurement of an arc in
Pennsylvania, extending more than one hundred miles.
[Illustration Fig. 12.]
[Illustration Fig. 13.]
These operations are carried on by what is called a system of
_triangulation_. Without some knowledge of trigonometry, you will not be
able fully to understand this process; but, as it is in its nature
somewhat curious, and is applied to various other geographical
measurements, as well as to the determination of arcs of the meridian, I
am desirous that you should understand its general principles. Let us
reflect, then, that it must be a matter of the greatest difficulty, to
execute with exactness the measurement of a line of any great length in
one continued direction on the earth's surface. Even if we select a
level and open country, more or less inequalities of surface will occur;
rivers must be crossed, morasses must be traversed, thickets must be
penetrated, and innumerable other obstacles must be surmounted; and
finally, every time we apply an artificial measure, as a rod, for
example, we obtain a result not absolutely perfect. Each error may
indeed be very small, but small errors, often repeated, may produce a
formidable aggregate. Now, one unacquainted with trigonometry can easily
understand the fact, that, when we know certain parts of a triangle, we
can find the other parts by calculation; as, in the rule of three in
arithmetic, we can obtain the fourth term of a proportion, from having
the first three terms given. Thus, in the triangle A B C, Fig. 12, if we
know the side A B, and the angles at A and B, we can find by
computation, the other sides, A C and B C, and the remaining angle at C.
Suppose, then, that in measuring an arc of the meridian through any
country, the line were to pass directly through A B, but the ground was
so obstructed between A and B, that we could not possibly carry our
measurement through it. We might then measure another line, as A C,
which was accessible, and with a compass take the bearing of B from the
points A and C, by which means we should learn the value of the angles
at A and C. From these data we might calculate, by the rules of
trigonometry, the exact length of the line A B. Perhaps the ground might
be so situated, that we could not reach the point B, by any route;
still, if it could be seen from A and C, it would be all we should want.
Thus, in conducting a trigonometrical survey of any country, conspicuous
signals are placed on elevated points, and the bearings of these are
taken from the extremities of a known line, called the base, and thus
the relative situation of various places is accurately determined. Were
we to undertake to run an exact north and south line through any
country, as New England, we should select, near one extremity, a spot of
ground favorable for actual measurement, as a level, unobstructed plain;
we should provide a measure whose length in feet and inches was
determined with the greatest possible precision, and should apply it
with the utmost care. We should thus obtain a _base line_. From the
extremities of this line, we should take (with some appropriate
instrument) the bearing of some signal at a greater or less distance,
and thus we should obtain one side and two angles of a triangle, from
which we could find, by the rules of trigonometry, either of the unknown
sides. Taking this as a new base, we might take the bearing of another
signal, still further on our way, and thus proceed to run the required
north and south line, without actually measuring any thing more than the
first, or base line. Thus, in Fig. 13, we wish to measure the distance
between the two points A and O, which are both on the same meridian, as
is known by their having the same longitude; but, on account of various
obstacles, it would be found very inconvenient to measure this line
directly, with a rod or chain, and even if we could do it, we could not
by this method obtain nearly so accurate a result, as we could by a
series of triangles, where, after the base line was measured, we should
have nothing else to measure except angles, which can be determined, by
observation, to a greater degree of exactness, than lines. We therefore,
in the first place, measure the base line, A B, with the utmost
precision. Then, taking the bearing of some signal at C from A and B, we
obtain the means of calculating the side B C, as has been already
explained. Taking B C as a new base, we proceed, in like manner, to
determine successively the sides C D, D E, and E F, and also A C, and C
E. Although A C is not in the direction of the meridian, but
considerably to the east of it, yet it is easy to find the corresponding
distance on the meridian, A M; and in the same manner we can find the
portions of the meridian M N and N O, corresponding respectively to C E
and E F. Adding these several parts of the meridian together, we obtain
the length of the arc from A to O, in miles; and by observations on the
north star, at each extremity of the arc, namely, at A and at O, we
could determine the difference of latitude between these two points.
Suppose, for example, that the distance between A and O is exactly five
degrees, and that the length of the intervening line is three hundred
and forty-seven miles; then, dividing the latter by the former number,
we find the length of a degree to be sixty-nine miles and four tenths.
To take, however, a few of the results actually obtained, they are as
follows:
Places of observation. Latitude. Length of a deg.
in miles.
Peru, 00° 00' 00" 68.732
Pennsylvania, 39 12 00 68.896
France, 46 12 00 69.054
England, 51 29 54-1/2 69.146
Sweden, 66 20 10 69.292
This comparison shows, that the length of a degree gradually increases,
as we proceed from the equator towards the pole. Combining the results
of various estimates, the dimensions of the terrestrial spheroid are
found to be as follows:
Equatorial diameter, 7925.648 miles.
Polar diameter, 7899.170 "
Average diameter, 7912.409 "
The difference between the greatest and the least is about twenty-six
and one half miles, which is about one two hundred and ninety-ninth part
of the greatest. This fraction is denominated the _ellipticity_ of the
earth,--being the excess of the equatorial over the polar diameter.
The operations, undertaken for the purpose of determining the figure of
the earth, have been conducted with the most refined exactness. At any
stage of the process, the length of the last side, as obtained by
calculation, may be actually measured in the same manner as the base
from which the series of triangles commenced. When thus measured, it is
called the _base of verification_. In some surveys, the base of
verification, when taken at a distance of four hundred miles from the
starting point, has not differed more than one foot from the same line,
as determined by calculation.
Another method of arriving at the exact figure of the earth is, by
observations with the _pendulum_. If a pendulum, like that of a clock,
be suspended, and the number of its vibrations per hour be counted, they
will be found to be different in different latitudes. A pendulum that
vibrates thirty-six hundred times per hour, at the equator, will vibrate
thirty-six hundred and five and two thirds times, at London, and a still
greater number of times nearer the north pole. Now, the vibrations of
the pendulum are produced by the force of gravity. Hence their
comparative number at different places is a measure of the relative
forces of gravity at those places. But when we know the relative forces
of gravity at different places, we know their relative distances from
the centre of the earth; because the nearer a place is to the centre of
the earth, the greater is the force of gravity. Suppose, for example, we
should count the number of vibrations of a pendulum at the equator, and
then carry it to the north pole, and count the number of vibrations made
there in the same time,--we should be able, from these two observations,
to estimate the relative forces of gravity at these two points; and,
having the relative forces of gravity, we can thence deduce their
relative distances from the centre of the earth, and thus obtain the
polar and equatorial diameters. Observations of this kind have been
taken with the greatest accuracy, in many places on the surface of the
earth, at various distances from each other, and they lead to the same
conclusions respecting the figure of the earth, as those derived from
measuring arcs of the meridian. It is pleasing thus to see a great
truth, and one apparently beyond the pale of human investigation,
reached by two routes entirely independent of each other. Nor, indeed,
are these the only proofs which have been discovered of the spheroidal
figure of the earth. In consequence of the accumulation of matter above
the equatorial regions of the earth, a body weighs less there than
towards the poles, being further removed from the centre of the earth.
The same accumulation of matter, by the force of attraction which it
exerts, causes slight inequalities in the motions of the moon; and since
the amount of these becomes a measure of the force which produces them,
astronomers are able, from these inequalities, to calculate the exact
quantity of the matter thus accumulated, and hence to determine the
figure of the earth. The result is not essentially different from that
obtained by the other methods. Finally, the shape of the earth's shadow
is altered, by its spheroidal figure,--a circumstance which affects the
time and duration of a lunar eclipse. All these different and
independent phenomena afford a pleasing example of the harmony of truth.
The known effects of the centrifugal force upon a body revolving on its
axis, like the earth, lead us to infer that the earth is of a spheroidal
figure; but if this be the fact, the pendulum ought to vibrate faster
near the pole than at the equator, because it would there be nearer the
centre of the earth. On trial, such is found to be the case. If, again,
there be such an accumulation of matter about the equatorial regions,
its effects ought to be visible in the motions of the moon, which it
would influence by its gravity; and there, also, its effects are traced.
At length, we apply our measures to the surface of the earth itself, and
find the same fact, which had thus been searched out among the hidden
things of Nature, here palpably exhibited before our eyes. Finally, on
estimating from these different sources, what the exact amount of the
compression at the poles must be, all bring out nearly one and the same
result. This truth, so harmonious in itself, takes along with it, and
establishes, a thousand other truths on which it rests.
FOOTNOTE:
[5] Library of Useful Knowledge: History of Astronomy, page 95.
LETTER VIII.
DIURNAL REVOLUTIONS.
"To some she taught the fabric of the sphere,
The changeful moon, the circuit of the stars,
The golden zones of heaven."--_Akenside._
WITH the elementary knowledge already acquired, you will now be able to
enter with pleasure and profit on the various interesting phenomena
dependent on the revolution of the earth on its axis and around the sun.
The apparent diurnal revolution of the heavenly bodies, from east to
west, is owing to the actual revolution of the earth on its own axis,
from west to east. If we conceive of a radius of the earth's equator
extended until it meets the concave sphere of the heavens, then, as the
earth revolves, the extremity of this line would trace out a curve on
the face of the sky; namely, the celestial equator. In curves parallel
to this, called the _circles of diurnal revolution_, the heavenly bodies
actually _appear_ to move, every star having its own peculiar circle.
After you have first rendered familiar the real motion of the earth from
west to east, you may then, without danger of misapprehension, adopt the
common language, that all the heavenly bodies revolve around the earth
once a day, from east to west, in circles parallel to the equator and to
each other.
I must remind you, that the time occupied by a star, in passing from any
point in the meridian until it comes round to the same point again, is
called a _sidereal day_, and measures the period of the earth's
revolution on its axis. If we watch the returns of the same star from
day to day, we shall find the intervals exactly equal to each other;
that is, _the sidereal days are all equal_. Whatever star we select for
the observation, the same result will be obtained. The stars, therefore,
always keep the same relative position, and have a common movement
round the earth,--a consequence that naturally flows from the hypothesis
that their _apparent_ motion is all produced by a single _real_ motion;
namely, that of the earth. The sun, moon, and planets, as well as the
fixed stars, revolve in like manner; but their returns to the meridian
are not, like those of the fixed stars, at exactly equal intervals.
The _appearances_ of the diurnal motions of the heavenly bodies are
different in different parts of the earth,--since every place has its
own horizon, and different horizons are variously inclined to each
other. Nothing in astronomy is more apt to mislead us, than the
obstinate habit of considering the horizon as a fixed and immutable
plane, and of referring every thing to it. We should contemplate the
earth as a huge globe, occupying a small portion of space, and encircled
on all sides, at an immense distance, by the starry sphere. We should
free our minds from their habitual proneness to consider one part of
space as naturally _up_ and another _down_, and view ourselves as
subject to a force (gravity) which binds us to the earth as truly as
though we were fastened to it by some invisible cords or wires, as the
needle attaches itself to all sides of a spherical loadstone. We should
dwell on this point, until it appears to us as truly up, in the
direction B B, C C, D D, when one is at B, C, D, respectively, as in the
direction A A, when he is at A, Fig. 14.
Let us now suppose the spectator viewing the diurnal revolutions from
several different positions on the earth. On the _equator_, his horizon
would pass through both poles; for the horizon cuts the celestial vault
at ninety degrees in every direction from the zenith of the spectator;
but the pole is likewise ninety degrees from his zenith, when he stands
on the equator; and consequently, the pole must be in the horizon. Here,
also, the celestial equator would coincide with the prime vertical,
being a great circle passing through the east and west points. Since all
the diurnal circles are parallel to the equator, consequently, they
would all, like the equator be perpendicular to the horizon. Such a
view of the heavenly bodies is called a right sphere, which may be thus
defined: _a right sphere is one in which all the daily revolutions of
the stars are in circles perpendicular to the horizon_.
[Illustration Fig. 14.]
A right sphere is seen only at the equator. Any star situated in the
celestial equator would appear to rise directly in the east, at midnight
to be in the zenith of the spectator, and to set directly in the west.
In proportion as stars are at a greater distance from the equator
towards the pole, they describe smaller and smaller circles, until, near
the pole, their motion is hardly perceptible.
If the spectator advances one degree from the equator towards the north
pole, his horizon reaches one degree beyond the pole of the earth, and
cuts the starry sphere one degree below the pole of the heavens, or
below the north star, if that be taken as the place of the pole. As he
moves onward towards the pole, his horizon continually reaches further
and further beyond it, until, when he comes to the pole of the earth,
and under the pole of the heavens, his horizon reaches on all sides to
the equator, and coincides with it. Moreover, since all the circles of
daily motion are parallel to the equator, they become, to the spectator
at the pole, parallel to the horizon. Or, _a parallel sphere is that in
which all the circles of daily motion are parallel to the horizon_.
To render this view of the heavens familiar, I would advise you to
follow round in mind a number of separate stars, in their diurnal
revolution, one near the horizon, one a few degrees above it, and a
third near the zenith. To one who stood upon the north pole, the stars
of the northern hemisphere would all be perpetually in view when not
obscured by clouds, or lost in the sun's light, and none of those of the
southern hemisphere would ever be seen. The sun would be constantly
above the horizon for six months in the year, and the remaining six
continually out of sight. That is, at the pole, the days and nights are
each six months long. The appearances at the south pole are similar to
those at the north.
A perfect parallel sphere can never be seen, except at one of the
poles,--a point which has never been actually reached by man; yet the
British discovery ships penetrated within a few degrees of the north
pole, and of course enjoyed the view of a sphere nearly parallel.
As the circles of daily motion are parallel to the horizon of the pole,
and perpendicular to that of the equator, so at all places between the
two, the diurnal motions are oblique to the horizon. This aspect of the
heavens constitutes an oblique sphere, which is thus defined: _an
oblique sphere is that in which the circles of daily motion are oblique
to the horizon_.